Relativity- energy

1. The problem statement, all variables and given/known data
<Most of this is irrelevant, the relevant parts are in italics>

The Greisen-Zatsepin-Kuzmin (GZK) upper limit on cosmic ray energies

The contemporary universe is filled with low energy photons left over from the time when the early universe cooled sufficiently for the omnipresent plasma of electrons and protons to form neutral hydrogen. These photons, born as visible- and ultraviolet-wavelength electromagnetic radiation, have lost energy as the universe has expanded. The current "cosmic microwave background" (CMB) spectrum is well described by a black body spectrum with temperature 2.725 K. This corresponds to a peak wavelength of 1.9 millimeters, or a peak energy per photon of about 6.5 x 10^-4 eV.

In 1966, Greisen, Kuzmin and Zatsepin realized that cosmic ray protons of sufficiently high energy could, in collisions with these photons, have their internal quark structure scrambled to produce heavier, unstable particles such as the Δ+ through the process γp → Δ+ → pπ0 and γp → Δ+ → nπ+. After the Δ+ decays, the proton (or neutron) would have significantly less energy in the "lab" frame than it had before the collision. (By lab frame I mean a frame in which the CMB spectrum is not Doppler shifted away from its peak wavelength of 1.9 mm.)

The rest masses of the proton and Δ+ are 0.93827 GeV/c2 and 1.232 GeV/c2 respectively.

At what proton energy should the GZK effect begin to make itself felt? You can assume that all CMB photons have energy 6.5x10-4eV.

2. Relevant equations
Energy = γmc2
Energy and momentum conservation.

3. The attempt at a solution
Since we're talking about the energy when the GZK effect BEGINS to be felt, Δ+ will be formed at rest, hence it's energy will be (1.232)GeV.

I know what the energy of the CMB photons are (6.5x10-4eV)

So, simply subtracting 6.5 x 10-4 eV from 1.232 GeV should give me the answer. But it doesn't, which makes sense because I didn't use relativity at all in this problem.

1. The problem statement, all variables and given/known data
<Most of this is irrelevant, the relevant parts are in italics>

The Greisen-Zatsepin-Kuzmin (GZK) upper limit on cosmic ray energies

The contemporary universe is filled with low energy photons left over from the time when the early universe cooled sufficiently for the omnipresent plasma of electrons and protons to form neutral hydrogen. These photons, born as visible- and ultraviolet-wavelength electromagnetic radiation, have lost energy as the universe has expanded. The current "cosmic microwave background" (CMB) spectrum is well described by a black body spectrum with temperature 2.725 K. This corresponds to a peak wavelength of 1.9 millimeters, or a peak energy per photon of about 6.5 x 10^-4 eV.

In 1966, Greisen, Kuzmin and Zatsepin realized that cosmic ray protons of sufficiently high energy could, in collisions with these photons, have their internal quark structure scrambled to produce heavier, unstable particles such as the Δ+ through the process γp → Δ+ → pπ0 and γp → Δ+ → nπ+. After the Δ+ decays, the proton (or neutron) would have significantly less energy in the "lab" frame than it had before the collision. (By lab frame I mean a frame in which the CMB spectrum is not Doppler shifted away from its peak wavelength of 1.9 mm.)

The rest masses of the proton and Δ+ are 0.93827 GeV/c2 and 1.232 GeV/c2 respectively.

At what proton energy should the GZK effect begin to make itself felt? You can assume that all CMB photons have energy 6.5x10-4eV.

2. Relevant equations
Energy = γmc2
Energy and momentum conservation.

3. The attempt at a solution
Since we're talking about the energy when the GZK effect BEGINS to be felt, Δ+ will be formed at rest, hence it's energy will be (1.232)GeV.

I know what the energy of the CMB photons are (6.5x10-4eV)

So, simply subtracting 6.5 x 10-4 eV from 1.232 GeV should give me the answer. But it doesn't, which makes sense because I didn't use relativity at all in this problem.